Canonical p-dimensions of algebraic groups and degrees of basic polynomial invariants

K. Zainoulline

September 22, 2005

Abstract In the present notes we provide a new uniform way to compute a canonical p-dimension of a split algebraic group G for a torsion prime p using degrees of basic polynomial invariants described by V. Kac. As an application, we compute the canonical p-dimensions for all split simple algebraic groups.

The notion of a canonical dimension of an algebraic structure was introduced by Berhuy and Reichstein [1]. For a split algebraic group G and its torsion prime p the canonical p-dimension of G was studied by Karpenko and Merkurjev in [3] and [4]. In particular, this invariant was shown to be related with the size of the image of the characteristic map

ϕ_{G} : S^{∗}(T̂) → CH^{∗}(X),  (1)

where T̂ is the character group of a maximal split torus T, X = G/B is the variety of complete flags and S^{∗} stands for the symmetric algebra. Namely, one has the following formula for the canonical p-dimension of a group G

cd_{p}(G) = min\{i | \overline{Ch_{i}}(X)\},  (2)

where \overline{Ch_{i}}(X) stands for the image of ϕ_{G} in the Chow ring Ch_{i}(X) = CH_{i}(X; ℤ/pℤ) with ℤ/pℤ-coefficients. Using this remarkable fact together with the explicit description of the image R_{p} = \overline{Ch}(X) Karpenko and Merkurjev computed the canonical p-dimensions for all classical algebraic groups G.

The goal of the present notes is to relate the work by V. Kac [2] devoted to the study of the p-torsion of the Chow ring of an algebraic group G with the canonical p-dimensions of G. In particular we provide a different and uniform approach of computing cd_{p}(G).

Theorem. Let G be a split simple algebraic group of rank n and p be a torsion prime. Then

cd_{p}(G) = N + n − (d_{1,p} + d_{2,p} + … + d_{n,p}),

where N stands for the number of positive roots of G and integers d_{1,p}, …, d_{1,n} are the degrees of basic polynomial invariants modulo p.

Proof. Consider the characteristic map (1) modulo p

(ϕ_{G})_{p} : S^{∗}(T̂) ⊗_{ℤ} ℤ/pℤ → Ch^{∗}(X)  (3)

According to [2, Theorem 1], the kernel of this map I_{p} is generated by a regular sequence of n homogeneous polynomials of degrees d_{1,p}, …, d_{n,p}.

Recall that a Poincare polynomial P (A, t) for a graded module M^{∗} over a field k is defined to be ∑_{i} dim_{k} M^{i} · t^{i} (see [5]). Hence, the Poincare polynomial for the ℤ/pℤ-module R_{p} is equal to

P (R_{p}, t) = ∏_{i=0}^{n} \frac{1 − t^{d_{i,p}}}{1 − t} .  (4)

Indeed, we identify R_{p} with the quotient of the polynomial ring in n variables ℤ/pℤ[ω_{1}, …, ω_{n}] modulo the ideal I_{p} (here ω_{i} are the fundamental weights). The formula (4) then follows immediately by [5, Cor. 3.3].

According to (2) the canonical p-dimension is equal to the difference dim(X) − deg P (R_{p}, t).

Corollary 1. We obtain the following values for the canonical p-dimensions of groups of types F_{4}, E_{6}, E_{7} and E_{8} (in the list below G^{sc} and G^{ad} denote the simply-connected and adjoint forms of G).

cd_{2} F_{4} = 3, cd_{3} F_{4} = 8
cd_{2} E_{6} = 3, cd_{3} E_{6}^{sc} = 8, cd_{3} E_{6}^{ad} = 16
cd_{2} E_{7}^{sc} = 17, cd_{2} E_{7}^{ad} = 18 cd_{3} E_{7} = 8
cd_{2} E_{8} = 60, cd_{3} E_{8} = 28 cd_{5} E_{8} = 24

Proof. Follows from the list of degrees of basic polynomial invariants provided in [2, Table 2].

Corollary 2. There is the following relation between the canonical p-dimension of G and the p-torsion part of the Chow group of G

cd_{p}(G) = max{i | Ch^{i}(G) ≠ 0}

Proof. It is known that Ch(G) = Ch(X)/J_{p}, where J_{p} is the ideal generated by the non-constant part of R_{p}. Since Ch(X) is a free R_{p}-module, we obtain P (Ch(X)/J_{p}, t) = P (Ch(X), t)/P (R_{p}, t).

Corollary 3. Let d_{1}, d_{2}, …, d_{n} be the degrees of “usual” basic polynomial invariants of G, i.e., with ℚ-coefficients. Then the canonical p-dimension of G is equal to

cd_{p}(G) = ∑_{i=0}^{n} d′_{i} · (p^{k_{i}} − 1),

where the integers d′_{i} and p^{k_{i}} are the factors of the decompositions d_{i} = d′_{i}·p^{k_{i}}, p ∤ d′_{i}. Observe that the degrees d_{i} for which k_{i} > 0 form the set of pexceptional degrees introduced in [2].

Proof. Follows by the isomorphism of [2, Theorem 3,(ii)].

Acknowledgements. I am grateful to Burt Totaro for mentioning to me the paper [2].

References

[1] Berhuy, G., Reichstein, Z. On the notion of canonical dimension for algebraic groups. Advances in Math., in press, doi: 10.1016/j.aim.2004.12.004.

[2] Kac, V. Torsion in cohomology of compact Lie groups and Chow rings of reductive algebraic groups. Invent. Math. 80 (1985), 69–79.

[3] Karpenko, N. Canonical dimension of orthogonal group. Transform. Groups 10 (2005), no. 2, 211–215.

[4] Karpenko, N., Merkurjev, A. Canonical p-dimension of algebraic groups. Advances in Math., in press, doi: 10.1016/j.aim.2005.07.013.

[5] Stanley, R. Hilbert functions of graded algebras. Advances in Math. 28 (1978), 57–83.